In this paper, we investigate stratification theory in terms of the
defining equations of strata and maps (without tube systems), offering
a concrete approach to show that some given family is topologically
trivial. In this approach, we consider a weighted version of
$(w)$-regularity condition and Kuo's ratio test condition.

There is a well-known correspondence (due to Mehta and Seshadri in
the unitary case, and extended by Bhosle and Ramanathan to other
groups), between the symplectic variety $M_h$ of representations of
the fundamental group of a punctured Riemann surface into a compact
connected Lie group~$G$, with fixed conjugacy classes $h$ at the
punctures, and a complex variety ${\cal M}_h$ of holomorphic bundles
on the unpunctured surface with a parabolic structure at the puncture
points. For $G = \SU(2)$, we build a symplectic variety $P$ of pairs
(representations of the fundamental group into $G$, ``weighted frame''
at the puncture points), and a corresponding complex variety ${\cal
P}$ of moduli of ``framed parabolic bundles'', which encompass
respectively all of the spaces $M_h$, ${\cal M}_h$, in the sense that
one can obtain $M_h$ from $P$ by symplectic reduction, and ${\cal
M}_h$ from ${\cal P}$ by a complex quotient. This allows us to
explain certain features of the toric geometry of the $\SU(2)$ moduli
spaces discussed by Jeffrey and Weitsman, by giving the actual toric
variety associated with their integrable system.

In this paper, we consider the (essential) spectrum of the discrete
Laplacian of an infinite graph. We introduce a new quantity for an
infinite graph, in terms of which we give new lower bound estimates of
the (essential) spectrum and give also upper bound estimates when the
infinite graph is bipartite. We give sharp estimates of the
(essential) spectrum for several examples of infinite graphs.

We develop a scattering theory for perturbations of powers of the
Laplacian on asymptotically Euclidean manifolds. The (absolute)
scattering matrix is shown to be a Fourier integral operator
associated to the geodesic flow at time $\pi$ on the boundary.
Furthermore, it is shown that on $\Real^n$ the asymptotics of certain
short-range perturbations of $\Delta^k$ can be recovered from the
scattering matrix at a finite number of energies.

Given a holomorphic Hilbertian bundle on a compact complex manifold, we
introduce the notion of holomorphic $L^2$ torsion, which lies in the
determinant line of the twisted $L^2$ Dolbeault cohomology and
represents a volume element there. Here we utilise the theory of
determinant lines of Hilbertian modules over finite von~Neumann
algebras as developed in \cite{CFM}. This specialises to the
Ray-Singer-Quillen holomorphic torsion in the finite dimensional case.
We compute a metric variation formula for the holomorphic $L^2$
torsion, which shows that it is {\it not\/} in general independent of
the choice of Hermitian metrics on the complex manifold and on the
holomorphic Hilbertian bundle, which are needed to define it. We
therefore initiate the theory of correspondences of determinant lines,
that enables us to define a relative holomorphic $L^2$ torsion for a
pair of flat Hilbertian bundles, which we prove is independent of the
choice of Hermitian metrics on the complex manifold and on the flat
Hilbertian bundles.

This paper treats the moduli space ${\cal M}_{g,1}(\Lambda)$ of
representations of the fundamental group of a Riemann surface of
genus $g$ with one boundary component which send the loop around
the boundary to an element conjugate to $\exp \Lambda$, where
$\Lambda$ is in the fundamental alcove of a Lie algebra. We
construct natural line bundles over ${\cal M}_{g,1} (\Lambda)$ and
exhibit natural homology cycles representing the Poincar\'e dual of
the first Chern class. We use these cycles to prove differential
equations satisfied by the symplectic volumes of these spaces.
Finally we give a bound on the degree of a nonvanishing element of
a particular subring of the cohomology of the moduli space of
stable bundles of coprime rank $k$ and degree $d$.

Let $\Gamma$ be a rank-one arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$-Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
square-integrable $\Gamma$-automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorge-Wallach
and Delorme.

I consider a two-parameter family $B_{s,t}$ of unitary transforms
mapping an $L^{2}$-space over a Lie group of compact type onto a
holomorphic $L^{2}$-space over the complexified group. These were
studied using infinite-dimensional analysis in joint work with
B.~Driver, but are treated here by finite-dimensional means. These
transforms interpolate between two previously known transforms, and
all should be thought of as generalizations of the classical
Segal-Bargmann transform. I consider also the limiting cases $s
\rightarrow \infty$ and $s \rightarrow t/2$.

We give necessary and sufficient conditions for a norm-compact subset
of a Hilbert space to admit a $C^1$ embedding into a finite dimensional
Euclidean space. Using quasibundles, we prove a structure theorem
saying that the stratum of $n$-dimensional points is contained in an
$n$-dimensional $C^1$ submanifold of the ambient Hilbert space. This
work sharpens and extends earlier results of G.~Glaeser on paratingents.
As byproducts we obtain smoothing theorems for compact subsets of
Hilbert space and disjunction theorems for locally compact subsets
of Euclidean space.

In this paper we establish conditions that guarantee, in the
setting of a general Banach space, the Painlev\'e-Kuratowski
convergence of the graphs of the subdifferentials of convexly
composite functions. We also provide applications to the
convergence of multipliers of families of constrained optimization
problems and to the generalized second-order derivability of
convexly composite functions.

We prove a uniform upper estimate on the number of cuspidal
eigenvalues of the $\Ga$-automorphic Laplacian below a given bound
when $\Ga$ varies in a family of congruence subgroups of a given
reductive linear algebraic group. Each $\Ga$ in the family is assumed
to contain a principal congruence subgroup whose index in $\Ga$ does
not exceed a fixed number. The bound we prove depends linearly on the
covolume of $\Ga$ and is deduced from the analogous result about the
cut-off Laplacian. The proof generalizes the heat-kernel method which
has been applied by Donnelly in the case of a fixed lattice~$\Ga$.

We develop a method of separable reduction for Fr\'{e}chet-like
normals and $\epsilon$-normals to arbitrary sets in general Banach
spaces. This method allows us to reduce certain problems involving
such normals in nonseparable spaces to the separable case. It is
particularly helpful in Asplund spaces where every separable subspace
admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable
reduction method in Asplund spaces, we provide a new direct proof of a
nonconvex extension of the celebrated Bishop-Phelps density theorem.
Moreover, in this way we establish new characterizations of Asplund
spaces in terms of $\epsilon$-normals.

This paper deals with periodic solutions for the billiard problem in a
bounded open set of $\hbox{\Bbbvii R}^N$ which are limits of regular
solutions of Lagrangian systems with a potential well. We give a
precise link between the Morse index of approximate solutions
(regarded as critical points of Lagrangian functionals) and the
properties of the bounce trajectory to which they converge.

We prove that all Liouville's tori generic bifurcations of a
large class of two degrees of freedom integrable Hamiltonian
systems (the so called Jacobi-Moser-Mumford systems) are
nondegenerate in the sense of Bott. Thus, for such systems,
Fomenko's theory~\cite{fom} can be applied (we give the example
of Gel'fand-Dikii's system). We also check the Bott property
for two interesting systems: the Lagrange top and the geodesic
flow on an ellipsoid.

Let $M(X,Y)$ denote the space of all continous functions
between $X$ and $Y$ and $M_f(X,Y)$ the path component
corresponding to a given map $f: X\rightarrow Y.$ When $X$ and
$Y$ are classical flag manifolds, we prove the components of
$M(X,Y)$ corresponding to ``simple'' maps $f$ are classified
up to rational homotopy type by the dimension of the kernel of
$f$ in degree two
cohomology. In fact, these components are themselves all products
of flag manifolds and odd spheres.

In this work we study the global geometry of planar quadratic
Hamiltonian systems with a center and we sum up the dynamics of
these systems in geometrical terms. For this we use the
algebro-geometric concept of multiplicity of intersection
$I_p(P,Q)$ of two complex projective curves $P(x,y,z) = 0$,
$Q(x,y,z) = 0$ at a point $p$ of the plane. This is a
convenient concept when studying polynomial systems and it
could be applied for the analysis of other classes of nonlinear
systems.

The spectral theory for the Neumann Laplacian on planar domains with
symmetric, horn-like ends is studied. For a large class of such domains,
it is proven that the Neumann Laplacian has no singular continuous
spectrum, and that the pure point spectrum consists of eigenvalues
of finite multiplicity which can accumulate only at $0$ or $\infty$.
The proof uses Mourre theory.

In \cite{Anker}, Jean-Philippe Anker conjectures an upper bound for the
heat kernel of a symmetric space of noncompact type. We show in this
paper that his prediction is verified for the space of positive
definite $n\times n$ real matrices.

In this paper, we study the local bifurcations of critical periods
in the neighborhood of a nondegenerate centre of the reduced Kukles
system. We find at the same time the isochronous systems. We show
that at most three local critical periods bifurcate from the
Christopher-Lloyd centres of finite order, at most
two from the linear isochrone and at most one critical period from the
nonlinear isochrone. Moreover, in all cases, there exist
perturbations which lead to the maximum number of critical
periods. We determine the isochrones, using the method of Darboux:
the linearizing transformation of an isochrone is derived from the
expression of the first integral.
Our approach is a combination of computational algebraic techniques
(Gr\"obner bases, theory of the resultant, Sturm's algorithm), the
theory of ideals of noetherian rings and the transversality theory
of algebraic curves.

In this paper we prove, that under certain hypotheses,
the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$,
$\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are
quasi-homogeneous vector fields, has at most two limit cycles.
The main tools used in the proof are the generalized polar
coordinates, introduced by Lyapunov to study the stability of degenerate
critical points, and the analysis of the derivatives of the Poincar\'e
return map. Our results generalize those obtained for polynomial
systems with homogeneous non-linearities.